Problem 32 - Calculus for JEE ADVANCED

Question

\(\begin{aligned} f^{\prime}(x) &=-\pi \sin \pi x+10+6 x+3 x^{2} \\\ &=3(x+1)^{2}+7-\pi \sin \pi x>0 \text { for all } x \end{aligned}\).

Step-by-Step Solution

Answer

The derivative is positive for all \( x \).

Step 1: Analyze the Given Expression

We start by analyzing the given derivative function: \[ f'(x) = -\pi \sin(\pi x) + 10 + 6x + 3x^2 \] This can be rewritten using the vertex form as: \[ f'(x) = 3(x+1)^2 + 7 - \pi \sin(\pi x) \] Our goal is to determine if this derivative is positive for all real numbers \( x \).

Step 2: Break Down Each Component

The expression has three components: - The quadratic term \( 3(x+1)^2 \), which is always non-negative for any real \( x \) since squares are non-negative.- The constant \( 7 \), which contributes a positive component.- The term \( -\pi \sin(\pi x) \), which oscillates between \(-\pi\) and \(\pi\).

Step 3: Evaluate the Range of Quadratic Term and Constant Combined

The quadratic term \( 3(x+1)^2 \) has a minimum value of zero. Adding the constant 7 gives a minimum of 7. Thus, the combination of \( 3(x+1)^2 + 7 \) results in a minimum of 7 for all \( x \).

Step 4: Consider the Influence of the Sine Term

The term \( -\pi \sin(\pi x) \) oscillates from \(-\pi\) to \(\pi\). Its maximum negative impact on the polynomial is \(-\pi\), and thus contributes as much as \(-\pi\) to the expression. Since \(\pi \approx 3.14\), it subtracts at most 3.14 from the polynomial.

Step 5: Assess If the Derivative is Always Positive

Given that the minimum value of the quadratic term and constant is 7, the maximum decrease from the sine term is 3.14, leading to a minimum value of \( 7 - 3.14 \approx 3.86 \). Thus, the derivative \( f'(x) \) is always positive since \( 3.86 > 0 \).

Key Concepts

Derivative analysis

In calculus, a derivative helps us understand how a function changes at any given point. It tells us the rate of change or slope of the function. For the expression provided, we are dealing with the derivative:

• \( f^{\prime}(x) = 3(x+1)^2 + 7 - \pi \sin(\pi x) \)

To know whether this function is always increasing or decreasing, we analyze if the derivative is positive or negative across all real \( x \). Here, we see a quadratic part, a constant term, and a trigonometric sine function, each contributing differently.
The goal is to ensure that this derivative remains positive for all values. This involves assessing the minimum value of each component. By analyzing the components systematically, we establish that our derivative is always greater than zero.

Quadratic expression

A quadratic expression is one of the simplest polynomial forms, and it defines a parabola when plotted on a graph. The expression \( 3(x+1)^2 \) in the derivative given is such a form.

• This expression is always greater than or equal to zero, regardless of the value of \( x \), due to properties of squaring a real number, which is always non-negative.
• Additionally, the coefficient '3' magnifies its effect, but the quadratic nature ensures it does not become negative.

Understanding this helps in knowing why this component has a minimum value of zero. The constant term \( +7 \) essentially shifts the parabola upwards, ensuring the lowest value of this quadratic expression plus constant is 7.

Trigonometric functions

Trigonometric functions, like sine in our expression, often oscillate between values. The term \( -\pi \sin(\pi x) \) represents such a function.

• The sine function alone oscillates between -1 and 1. When multiplied by \(-\pi\), this becomes \(-\pi\) to \(\pi\).
• These oscillations can affect the sum in periodic intervals.

The most significant impact here is that the \(-\pi\) shifts the entire derivative downward by up to \(3.14\). Therefore, it's critical to assess if the combined effect of quadratic plus constant outweighs these negative oscillations.

Inequalities

Inequalities involve comparing values to determine their relative sizes. We use them here to figure if the entire expression is greater than zero for all \( x \).

• By establishing that the quadratic and constant sum up to at least 7, we determine the minimum positive value.
• Subtracting at most \(3.14\) from the sine function results, we assess whether the inequality \(7 - 3.14 > 0\) holds true.

This approach ensures the inequality confirms the positivity. By ensuring the derivative remains positive through these principles, we accomplish our goal. This positivity means the function is continuously increasing over real numbers without exception.